What does it mean for an object to not be following a definition based on some implication

Question

I want to get a deeper understanding of what being an object that doesn't follow a definition means in terms of predicates and logical operators.

Suppose the following definition of the closedness of a subset of states $C$ from a set of states $E$:

$C$ is closed means that(I guess you can think of "means that" as a logical equivalence)

if "$i \in C \land i,j \in E \land i\to j$" then "$j \in C$", which can be rewritten as

"$i \in C \land i,j \in E \land i\to j$" $\implies$ "$j \in C$"

Note: $\to$ is meant here as some property between two states

then how could you define the fact that $C$ isn't closed in terms of logical operators: $\in, \land, \lnot, \lor, \implies $...

Answer 1

The full defintion is :

"$C$ is closed iff for all $i,j ∈ E$, if $i∈C$ and $i→j$, then $j∈C$".

The negation of $P ⇒ Q$ is $P ∧ ¬Q$. Thus, we have:

"$C$ is not closed iff there are $i,j ∈ E$ such that: $i∈C$ and $i→j$ and $j∉C$".